Let A be a square n-by-n matrix. A scalar 
is called
an eigenvalue and a non-zero column vector v the corresponding
right eigenvector if 
. A nonzero column vector u
satisfying 
is called the left eigenvector .
One variant of the nonsymmetric eigenvalue problem requires the computation
of all n values of and,
if desired, their associated right eigenvectors v and/or
left eigenvectors u.
A second is the computation of the Schur factorization of a matrix A.
If A is complex, then its Schur factorization is 
, where
Z is unitary and T is upper triangular. If A is real, its
Schur factorization is 
, where Z is orthogonal.
and T is upper quasi-triangular (1-by-1 and 2-by-2 blocks on
its diagonal).
The columns of Z are called the Schur vectors of A.
The eigenvalues of A appear on the diagonal of T; complex conjugate
eigenvalues of a real A correspond to 2-by-2 blocks on the diagonal of T.
For more details, see for example, [55] .